870 research outputs found
Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field
Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local 'hot spot' on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Peclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity held further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Peclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Peclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field
Functional Renormalization for Disordered Systems, Basic Recipes and Gourmet Dishes
We give a pedagogical introduction into the functional renormalization group
treatment of disordered systems. After a review of its phenomenology, we show
why in the context of disordered systems a functional renormalization group
treatment is necessary, contrary to pure systems, where renormalization of a
single coupling constant is sufficient. This leads to a disorder distribution,
which after a finite renormalization becomes non-analytic, thus overcoming the
predictions of the seemingly exact dimensional reduction. We discuss, how the
non-analyticity can be measured in a simulation or experiment. We then
construct a renormalizable field theory beyond leading order. We discuss an
elastic manifold embedded in N dimensions, and give the exact solution for N to
infinity. This is compared to predictions of the Gaussian replica variational
ansatz, using replica symmetry breaking. We further consider random field
magnets, and supersymmetry. We finally discuss depinning, both isotropic and
anisotropic, and universal scaling function.Comment: 29 page
Slim Fractals: The Geometry of Doubly Transient Chaos
Traditional studies of chaos in conservative and driven dissipative systems
have established a correspondence between sensitive dependence on initial
conditions and fractal basin boundaries, but much less is known about the
relation between geometry and dynamics in undriven dissipative systems. These
systems can exhibit a prevalent form of complex dynamics, dubbed doubly
transient chaos because not only typical trajectories but also the (otherwise
invariant) chaotic saddles are transient. This property, along with a manifest
lack of scale invariance, has hindered the study of the geometric properties of
basin boundaries in these systems--most remarkably, the very question of
whether they are fractal across all scales has yet to be answered. Here we
derive a general dynamical condition that answers this question, which we use
to demonstrate that the basin boundaries can indeed form a true fractal; in
fact, they do so generically in a broad class of transiently chaotic undriven
dissipative systems. Using physical examples, we demonstrate that the
boundaries typically form a slim fractal, which we define as a set whose
dimension at a given resolution decreases when the resolution is increased. To
properly characterize such sets, we introduce the notion of equivalent
dimension for quantifying their relation with sensitive dependence on initial
conditions at all scales. We show that slim fractal boundaries can exhibit
complex geometry even when they do not form a true fractal and fractal scaling
is observed only above a certain length scale at each boundary point. Thus, our
results reveal slim fractals as a geometrical hallmark of transient chaos in
undriven dissipative systems.Comment: 13 pages, 9 figures, proof corrections implemente
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